I don't know the answer for sure, though I do not remember there being anything in the technical or sporting regulations requiring that cars be painted.
But I'd speculate that regardless of if cars are required to be painted or not in modern F1, that it would be done anyways to smooth the surface of the car for lower drag.
The weight it adds doesnt matter since all teams are adding ballast to the cars anyways to meet the minimum weight requirement of 605kg dry. Though one might argue that without the added weight of paint, more ballast could be added to have greater effect on the car balance.
I'm just quickly reading a few sections of the sporting and technical regulations now to see if there's any mention of paint.
edit:
from Sporting regulations
CAR LIVERY
60) The provisions of the Code relating to national colours shall not apply to the Championship. Both cars entered by a competitor must be presented in substantially the same livery at each Event, any change to this livery during a Championship season may only be made with the agreement of the Formula One Commission.
In order that the cars of each team may be easily distinguished from one another whilst they are on the track, the on board cameras located above the principle roll structure of the first car must be predominantly fluorescent red and the second car fluorescent yellow.
61) Each car will carry the race number of its driver (or his replacement) as published by the FIA at the beginning of the season. This number must be clearly visible from the front of the car.
62) The name or the emblem of the make of the car must appear on the front of the nose of the car and in either case be at least 25mm in its largest dimension. The name of the driver must appear on the external bodywork and be clearly legible.
So, nothing there saying the car has to be painted, apart from the red/yellow on the roll structure, and decals for driver name and number.
Everyone should be familiar with regulation 60 from the dispute in 1999 about BAR's paint schemes when they wanted to have Villeneuve and Zonta racing with different liveries on their cars. (Villeneuve's car was to have the Lucky Strike livery, and Zonta's was to have 555 if I remember everything correctly)